Cohomological Tensor Functors on Representations of the General Linear Supergroup

Abstract

We define and study cohomological tensor functors from the category T n of finite-dimensional representations of the supergroup Gl(n|n) into T n−r for 0 < r ≤ n. In the case DS : T n → T n−1 we prove a formula DS(L) = Π ni L i for the image of an arbitrary irreducible representation. In particular DS(L) is semisimple and multiplicity free. We derive a few applications of this theorem such as the degeneration of certain spectral sequences and a formula for the modified superdimension of an irreducible representation.

Duflo-Serganova functor and superdimension formula for the periplectic Lie superalgebra

“…Remark 1.2.1. A similar result for the general linear Lie superalgebra was proved in [HW14] using a similar technique. However, in contrast with the gl(m|n)-case, DS x (L n (λ)) may be not semisimple.…”

A few remarks on symmetries of the Duflo-Serganova functor

“…We will connect this computation of [HW14] to our approach, showing that one can view their computation as computing the action of the enveloping algebra of a Lie supergroup that arises naturally from the Duflo-Serganova functor. Further, the technique of Heidersdorf and Weissauer may also be applied to the so-called thin and thick Kacmodules of the periplectic Lie superalgebra.…”

“…Finally, the third approach is discussed in section 5; namely we produce an associative superalgebra which is a subquotient of Ug and acts on DS u . As of yet we do not understand how to compute this superalgebra in most cases, but we introduce it because it generalizes, in some sense, the construction given in [HW14]. We explain this connection, and further generalize it to the case of p(n); in the process we compute DS u on thin and thick Kac-modules when u is a certain maximal rank element.…”

Mixed tensors of the general linear supergroup

“…Then L(λ † ) ∨ = L( λ † ). For an explicit description of the dual of an irreducible GL(n|n)-module using this see [HW14].…”